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  • Too Risk Averse to Purchase Insurance? A Theoretical Glance at the Annuity Puzzle

    Antoine Bommier, Francois Le Grand

    ETH Risk Center – Working Paper Series

    ETH-RC-12-002

    The ETH Risk Center, established at ETH Zurich (Switzerland) in 2011, aims to develop cross- disciplinary approaches to integrative risk management. The center combines competences from the natural, engineering, social, economic and political sciences. By integrating modeling and simulation efforts with empirical and experimental methods, the Center helps societies to better manage risk.

    More information can be found at: http://www.riskcenter.ethz.ch/.

  • ETH-RC-12-002

    Too Risk Averse to Purchase Insurance? A Theoretical Glance at the Annuity Puzzle

    Antoine Bommier, Francois Le Grand

    Abstract

    This paper suggests a new explanation for the low level of annuitization, which is valid even if one as- sumes perfect markets. We show that, as soon there exists a positive bequest motive, sufficiently risk averse individuals should not purchase annuities. A model calibration accounting for temporal risk aver- sion generates a willingness-to-pay for annuities, which is significantly smaller than the one generated by a standard Yaari (1965) model. Moreover, the calibration predicts that riskless savings finances one third of consumption, in line with empirical findings. Keywords: annuity puzzle, insurance demand, bequest, intergenerational transfers, temporal risk aversion, multiplicative preferences.

    Keywords: annuity puzzle, insurance demand, bequest, intergenerational transfers, temporal risk aver- sion, multiplicative preferences

    Classifications: JEL Codes: D11, D81, D91.

    URL: http://web.sg.ethz.ch/ethz risk center wps/ETH-RC-12-002

    Notes and Comments:

    ETH Risk Center – Working Paper Series

  • Too Risk Averse to Purchase Insurance?

    A Theoretical Glance at the Annuity Puzzle∗

    Antoine Bommier François Le Grand†

    January 23, 2012

    Abstract

    This paper suggests a new explanation for the low level of annuitization, which is valid

    even if one assumes perfect markets. We show that, as soon there exists a positive bequest

    motive, sufficiently risk averse individuals should not purchase annuities. A model calibration

    accounting for temporal risk aversion generates a willingness-to-pay for annuities, which is

    significantly smaller than the one generated by a standard Yaari (1965) model. Moreover,

    the calibration predicts that riskless savings finances one third of consumption, in line with

    empirical findings.

    Keywords: annuity puzzle, insurance demand, bequest, intergenerational transfers, tem-

    poral risk aversion, multiplicative preferences.

    JEL codes: D11, D81, D91.

    1 Introduction

    Among the greatest risks in life is that associated with life expectancy. A recently retired American

    man of age 65 has a life expectancy of about 17.5 years. Though, there is a more than 22% chance

    that he will die within the first 10 years and a more than 20% chance that he will live more than

    25 years. Savings required to sustain 10 or 25 years of retirement vary considerably, and one would

    expect a strong demand for annuities, which are financial securities designed to deal with such a ∗We are grateful to Edmund Cannon, Alexis Direr, Lee Lockwood, Thomas Post, James Poterba, Ray Rees,

    Harris Schlesinger and seminar participants at ETH Zurich, University of Paris I, University of Zurich, 2011 Summer

    Meetings of the Econometric Society for their comments. †Antoine Bommier is Professor at ETH Zurich; abommier@ethz.ch François Le Grand is Assistant Professor at

    EMLyon Business School and Associate Researcher at ETH Zurich; legrand@em-lyon.com. Both authors gratefully

    acknowledge financial support from Swiss-Re.

    1

  • lifetime uncertainty. A number of papers have stressed the utility gains that would be generated

    by the annuitization of wealth at retirement. It is generally estimated that individuals would be

    willing to give up to 25% of their wealth at retirement to gain access to a perfect annuity market

    (see Mitchell, Poterba, Warshawsky and Brown (1999) among others). According to standard

    theoretical predictions, even when individuals have a bequest motive, they should fully annuitize

    the expected value of their future consumption. However, puzzlingly enough, empirical evidence

    consistently shows that individuals purchase very few private annuities, in sharp contradiction

    with the theoretical predictions. For example, Johnson, Burman, and Kobes (2004) report than

    in the US, private annuities finance less than 1% of household income for people older than 65.1

    Similarly, they also observe that private annuities are only purchased by 5% of people older than

    65. James and Song (2001) find similar results for other countries, such as Canada, the United

    Kingdom, Switzerland, Australia, Israel, Chile and Singapore.

    A number of explanations to this puzzle have been suggested, relying on market imperfections

    or rationality biases.2 For example, due to imperfect health insurance, individuals would need

    to store a substantial amount of liquidities; unfair annuity pricing would make them unattractive

    assets; or framing effects would play an important role in agents’ decisions to annuitize.

    In this paper we emphasize that, even if the annuity market were perfect, a low (or even zero)

    level of annuitization can be fully rational. Our explanation relies on the role of risk aversion.

    We show that a high level of risk aversion together with a positive bequest motive is sufficient to

    predict a negative demand for annuities.

    The reason why this explanation remained unexplored is that the literature has mainly focused

    on time additively separable preferences, or on Epstein and Zin specification, while both models

    are unadapted to study the role of risk aversion (See Bommier, Chassagnon, LeGrand (2010),

    henceforth BCL). In the current paper, the role of risk aversion is investigated in the expected

    utility framework, through the concavification of the lifetime utility function as introduced by

    Kihlstrom and Mirman (1974). We prove that the demand for annuities decreases with risk aversion

    and eventually vanishes when risk aversion is large enough.

    The fact that annuity demand decreases – and does not increase – with risk aversion might

    seem counterintuitive. Insurance demand is generally found to increase with risk aversion. Though,

    this correlation does not hold when irreplaceable commodities, such as life, are at risk. As was

    explained by Cook and Graham (1977), rational insurance decisions aim at equalizing marginal

    utilities of wealth across states of nature. With irreplaceable commodities, this may generate risk 1Roughly one half of income stems from public pensions, 17% from firm sponsored pension payments and one

    third is financed from savings. 2See Brown (2007), as well as the following section for a literature review.

    2

  • taking behavior. Whenever this is the case, one should expect risk aversion to decrease risk taking

    and reduce the demand for insurance.3

    Annuities provide an example where purchasing insurance is risk increasing. Lifetime is un-

    certain, but living long is generally considered to be a good outcome, while dying early is seen

    to be a bad outcome. For a given amount of savings, purchasing annuities, rather than bonds

    for example, involves reducing bequest in the case of an early death (i.e., a bad outcome), while

    increasing consumption in case of survival (i.e., a good outcome). Thus, for a given level of savings,

    annuities transfer resources from bad to good states of the world and are, as such, risk increasing.

    If first period consumption were exogenous and inter vivos transfers ruled out, simple dominance

    arguments as in BCL would directly imply that the demand for annuity decreases with risk aver-

    sion. In the current paper, the result is obtained with endogenous consumption smoothing and the

    introduction of inter vivos transfers. Moreover, we prove that when risk aversion is large enough,

    annuity demand eventually vanishes.

    In order to evaluate to what extent risk aversion contributes to solving the annuity puzzle, we

    calibrate a life-cycle model in which agents can invest in bonds and annuities. Calibrating risk

    aversion and bequest motives to plausible levels shows that risk aversion alone does not generate

    a negative demand for annuities. However, we obtain considerably smaller willingnesses-to-pay

    for annuities than those obtained with the standard Yaari model, indicating that risk aversion

    may indeed be an important factor to explain the low levels of annuitization. Our calibration

    implies that one third of the agents’ consumption is financed by riskless savings, which is in line

    with empirical findings of Johnson, Burman, and Kobes (2004). This contrasts with the standard

    Yaari’s model in which riskless savings do not contribute at all to consumption financing, even if

    agents have bequest motives.

    The remainder of the paper is structured as follows. In Section 2, we discuss the related

    literature. We then present a two-period model and derive our theoretical predictions in Section 3.